Solving Quadratics By Factoring Part I Factoring a = 1

Ways to factor… There a few different ways to approaching factoring an expression. However, the first thing you should always look for is the Greatest Common Factor (GCF). How to find the GCF video Factoring using GCF video

Why GCF first? You look for the GCF first because it will help you factor quadratics using the second method by making the numbers smaller. The second method of factor involves undoing the distributive property….I call it unFOILing  There are several videos posted on this method, but here’s an example…..

Factor: x2 + 6x + 8 Look at the last number. If the sign in a positive, the signs in the parenthesis will be the same. x2 + 6x + 8 Here the 8 is positive. Look at the sign on the middle number. We know the signs will be the same because 8 is positive. We look a the middle number and it's also positive. So both signs in the parenthesis will be positive.  (x +  )(x +  ) Find factors of the last number that when you mulitply them you get that last number, but when you combine them you get the middle number. So we're looking for factors of 8 that we multiply them we get an 8, but when we add them we get a 6.....4 and 2. (x + 4)(x + 2) Check it with FOIL. You never get a factoring problem wrong! You can always check it by multiplying. (x + 4)(x + 2) = x2 + 4x +2x +8                                            It works!

Factor: x2 - 7x + 12 Look at the last number. If the sign in a positive, the signs in the parenthesis will be the same. x2 - 7x + 12 Here the 12 is positive. Look at the sign on the middle number. We know the signs will be the same because 12 is positive. We look a the middle number and it's negative. So both signs in the parenthesis will be negative.  (x -  )(x -  ) Find factors of the last number that when you mulitply them you get that last number, but when you combine them you get the middle number. So we're looking for factors of 12 that we multiply them we get an 12, but when we add them we get a 7.....4 and 3. (x - 4)(x - 3) Check it with FOIL. You never get a factoring problem wrong! You can always check it by multiplying. (x - 4)(x - 3) = x2 - 4x - 3x +  12                       = x2 - 7x + 12                                                             

Factor: x2 - 3x - 54 Look at the last number. If the sign in a negative, the signs in the parenthesis will be opposites. x2 - 3x - 54 Here the 54 is negative. Look at the sign on the middle number. Since we know we will have one of each, the middle number now gives the sign to the greater factor.  (x -  )(x +  ) Find factors of the last number that when you mulitply them you get that last number, but when you combine them you get the middle number. So we're looking for factors of 54 that we multiply them to a get a 54, but when we subtract them we get a 3.....9 and 6. (x - 9)(x + 6) Check it with FOIL. You never get a factoring problem wrong! You can always check it by multiplying. (x - 9)(x + 6) = x2 - 9x + 6x - 54                       = x2 - 3x - 54

Special Case: The Difference of two Perfect Squares The difference of two perfect squares is very easy to factor, but everyone always forgets about them.!They're in the form (ax)2 - c where a and c are perfect squares. There's no visible b-value...so b = 0. You factor them by taking the square root of a and the square root of c and placing them in parenthesis that have opposite signs. Whenever you have a binomial that is subtraction, always check to see it’s this special case. It usually does NOT have a GCF. Here's an example….

Difference of Two Perfect Squares Video Example Factor : 4x2 – 9 Set up parenthesis with opposite signs (   +   )(   -   ) Find the square root of a and place then answer in the front sections of the parenthesis sqrt(4x2) =  2x ( 2x  +   )( 2x   -   ) Find the square root of c and place them at the end of the parenthesis. sqrt(9) = 3 ( 2x  + 3 )( 2x   - 3 ) Difference of Two Perfect Squares Video

Practice Factoring 1. x2 + 4x – 5 2. x2 - 3x + 2 3. x2 - 6x – 7

Solutions 1. x2 + 4x – 5 = (x+5)(x-1) 2. x2 - 3x + 2 = (x-1)(x-2)

Practice: Common Factors Practice: Difference of Two Squares Practice: Factor a = 1 Factoring with Algebra Tiles

Solving Quadratics By Factoring Part II Factoring a=1

What if the leading coefficient isn’t a 1? Factor: 3x2 + 11x - 4  Set up two pairs of parenthesis (          )(         ) Look over the equation (    +    )(    -   ) Look at the a-value Unfortunately, the a-value is not a one, so we need to list factors in a chart. We're looking for the pair of factors that when I find the difference of the products will yield the b-value.   Factors of A     Factors of C                      1, 3        2,2 and 1,4       1*2 - 3*2 = -4  NO           1*3 - 1*4= -1 NO       1*1 - 3*4=  -12  YES!  Enter in values (x - 4)(3x + 1) Check with FOIL It's possible that you have the right numbers but in the wrong spots, so you have to check. (x - 4)(3x + 1)= 3x^2 -12x + x - 4                        = 3x^2 -11x -4

Factoring when a≠ 1 Terms in a quadratic expression may have some common factors before you break them down into linear factors. Remember, the greatest common factor, GCF, is the greatest number that is a factor of all terms in the expression. When a ≠ 1, we should always check to see if the quadratic expression has a greatest common factor.

Therefore, 2x2 -22x +36 is = 2 (x -2)(x-9). Factor 2x2 -22x +36 Step 1: a ≠ 1, so we should check to see if the quadratic expression has a greatest common factor. It has a GCF of 2. 2x2 -22x +36 = 2(x2 -11x +18) Step 2: Once we factor out the GCF, the quadratic expression now has a value of a =1 and we can use the process we just went through in the previous examples. x2 -11x +18 = (x -2)(x-9) Therefore, 2x2 -22x +36 is = 2 (x -2)(x-9).

A≠ 1 and NO GCF 2x2 + 13x – 7 Step 1: a ≠ 1, so we should check to see if the quadratic expression has a greatest common factor. It does not have a GCF! This type of trinomial is much more difficult to factor than the previous. Instead of factoring the c value alone, one has to also factor the a value. Our factors of a become coefficients of our x-terms and the factors of c will go right where they did in the previous examples.

2x2 + 13x – 7 Step 1: Find the product ac. ac= -14 Step 2: Find two factors of ac that add to give b. 􀀹 1 and -14 = -13 􀀹 -1 and 14 = 13 This is our winner! 􀀹 2 and -7 = -5 􀀹 -2 and 7 = 5 Step 3: Split the middle term into two terms, using the numbers found in step above. 2x2 -1x + 14x – 7

Find the GCF for each column and row! Step 4: Factor out the common binomial using the box method. 2x2 -1x + 14x – 7 Quadratic Term Factor 1 Factor 2 Constant Term 2x2 -1x 14x -7 Find the GCF for each column and row!

Numbers in RED represent the GCF of each row and column 2x -1 x 2x2 -1x 7 14x -7 The factors are (x + 7)(2x - 1).

Practice Factoring 1. 2x2 11x + 5 2. 3x2 - 5x - 2 3. 7x2 - 16x + 4

Solutions 1. 2x2 +11x + 5 = (2x + 1)(x + 5)

Special Products Difference of Squares x2 - y2 = (x - y) (x + y) Square of Sum x2 + 2xy + y2 = (x + y)2 Square of Difference x2 - 2xy +y2= (x - y)2

Factoring Strategies 2 Terms… Is there a GCF? Look for special products. 3 Terms… Look for squares of a difference or a sum.

Prime Factors Remember: This won’t work for all quadratic trinomials, because not all quadratic trinomials can be factored into products of binomials with integer coefficients. We call these prime! (Prime Numbers are 3, 5, 7, 11, 13, etc.) Expressions such as x2 + 2x - 7, cannot be factored at all, and is therefore known as a prime polynomial.

Practicing Factoring when a ≠1. Please watch the demonstration below on factoring when a ≠ 1. There will be interactive examples provided to help when a ≠ 1. MORE FACTORING Upon completion of the video and demonstration, please complete Mastery Assignment Part 2.

More Instruction Gizmo: Factoring ax2 + bx + c Practice: All Other Cases Practice: Application Problems More Instruction